Estimation of the functional A_2 • A_3 In the class of bounded symmetric univalent functions

A C T A U N I V E R S I T A T I S L O D Z I E N S I S FOLIA MATHEMATICA 2, 1987

Longin Pietrasik

ESTIMATION OF THE FUNCTIONAL A2 • A?

IN THE CLASS OF BOUNOED SYMMETRIC UNIVALENT FUNCTIONS

Denote by Sp(M), M > 1, the family of functions F(z) = z + + Y. Anz0 holomorphic and univalent in the disc E = { z : | z | < l}, with

n=2

real coefficients and satisfying the condition |F(z)|SM, z e E. In the paper there has been obtained a sharp estima

In the paper there has been obtained a sharp estimation of the func­ tional H(F) = Aj* Aj in the classes Sp(M), M > 1.

Introduction

Let SR stand for the family of functions 00 (0.1) F (z) » z + £ Anzn

n=2

holomorphic and univalent in the disc E = -[z ; | z |< l } , with real coefficients.

Denote by S^iM), M > 1, the subfamily of the former, con­ sisting of bounded functions, i.e. of those satisfying the con­ dition

|F(z)| S M for z 6 E

The problems connected with the estimation of coefficients in the classes defined above were dealt with by many mathematicians.

In the family SR (M) well known are, among others, the follow­ ing sharp estimations:

1) for each function F e SR (M) [5],

(0.2) |A2 I S 2(1 - ^) for M > 1; 2) for each function F e SR (M) ([7], [9], [3]),

(0.3) |Aj|

1 - M-2 when 1 < M £ e

1 ♦ 2X2 - 4AM" 1 + M~ 2 when e S M < +oo

where the X occurring in (0.3) in the greater root of the equation X l g X = -M* 1

3) for each function F e SR (M), n even and M sufficiently large [8], [2], [1 0], [ll]),

(0.4) A :£ P-(M)

ll n . ^

where Pn (M) is the n-th coefficient in the Maclaurin expansion of a Pick function w = M) defined by the equation

(0.5) - - - *— - = — — ?- - - z e E, <>(0, M) » 0 (1 - ft) 2 (1 - z)2 ,

Moreover, this function realizes the equality in estimation

( 0 . 2 ) . ■ v

From estimation (0.3) it follows that the Pick function does not realize the maximum of the functional H(F) = A ^ , F e SR (M), M > 1.

The above-mentioned results justify the purposefulness of the investigation of the functional of the form

(0.6) H(F) = A2 • Aj

defined in the family SR (M). It is worth noting that (0.6) is not

a linear functional. i'!

To estimate the functional H(F) = A2 • Ay, the variational me­ thod is made use of in the paper. In particular, we use the gene­ ral equation for extremal functions, obtained by I . 0 z i u- b i r i s k i [l].

1. The equation for functions extremal with res poet to the Jfunctional A2 • A,

defined in the class S„(M)

Consider the functional

(1.1) H(F) = A2 • A3

defined on the family SR (M), M > 1. Since functional (1.1) is continuous, and the family SR (M) compact, there exists a function F* e Sr(M) of the form

(1.2) w = F*(z) = z A»zn

n-2

for which the functional attains its maximum. It is easy to notice that H(F*) > 0 and, thereby, A£ * 0 and A^ 1 0. Moreover, in virtue of [l], each of the extremal functions satisfies the following differential-functional equation:

r- # -* 2 ( 1 . 3 ) T il ( F * ( z ) ) e. i » . » ( z ) , 0 < I z I < 1 L F * ( z ) . z 1 where W ( w > 1 E D£-i[(S) P' 1 ♦ < m) 1' p1 - p = 2 3 w(z) r Ep - i ( z P l + z l P ) ■ 2V* P=1

DJ = 2M"2(A^ + 2AJ)', D* 2 a 2M'3AJ (1.4) E* = 3M*1A5A5, EJ = 2M'1[a^ ♦ 2A|2]

E-> * 2M-1AS, V * - min [0i cos x ♦ OS cos 2x]

The functions 7Jl(w) and 7 1 (2) take real-negative values on the circles IwI = M and |z| = 1, respectively, and either of them has at least one double zero on the respective circle just mention­ ed. It can easily be observed that if is a zero of the

func-2° 2

M M

tion 7TC(w), so are the numbers wQ , — -, analogously if z

0 0 1 1

is a zero of the function

71

(z), so are the numbers zQ , y

~.

Let us denote 0 0

M(AS + 2AS)

Cl.5) -- V — ^

2

Then formulae (1.4) will take the following form:

(1.6) 7n( ( 1 . 7 ) 71 ( z ) = 2AS [w* + |3Mw3 - - 4t~w2 + (3M3w + M4 ]

w, . —1L--- --- ,-- 1

mV 2A5 [z* , f , ] - V » , , j M z2

(1.8)

V*

--^»min AS [cos 2x + |3cos x] M Goc<2fl

Lemma 1. Each function (1.2) extremal with respect to functio­ nal (1.1) satisfies the equation:

I . The case A£ > 0. A. if 0<(3<;4; 8 . r z v'\2 A 2 (w2 +łpMw + m2>2 A2 (z2 + IS 2 + 1)2 ^ ” 4 2 “ 2 M w ZL

.

-2 AS(w + M ) 2 [w2 ♦ (fi - 2) Mw + M2J p j j -AS(z2 + A z + I)2

L

2M -

---if 4 < (3 < 4 M; C. if ß ä 4M. II. Th( 0. 1Î ß <0; E. if ß - 0, F, AJ (w + M) 2 [w2 ♦ (ß - 2) Mw + M2] M4w2 AJ(z ♦ l) 2 [z2 ♦ ( £ - 2)z ♦ l] z2 < 0. AJ (w M) 2 [ w2 + M ( ß - 2)w + M 2 ] M4w2 AJ(z ♦ ₁)2 [:z2 +(-§• - 2)z + l] z2 ( F - f AJ(w - M) 2 (w + M) 2 mV i > |N>* (z + l) 2 ( z - l) 2 z2 AJ (w - [w2 + M ( 0 + 2)w + M] 2 M*w2 AJ(z - l)2[z2 +(£ + 2) z l]

Besides,

(1.9) H(F*)=

in case A 2

y A^Tl + ( -ig-- (3 + 1)J in case B M 7 A2 [ ^ ł + M *

l]

Ą AJ(1 - (1 + - f) A 5 (1 ” "^j) M 3 2 -y A5(1 * ¥ ) ( 1 + ¥ + m5 in case C in case D in case E in case F

P r o o f . Consider case I. Since > 0 and A J ^ A ^ X ) , the­ refore from (1.5) it follows that, in this case, (3 > 0 .

Let us calculate V * . For the purpose, put u(x) = (3 cos x t cos 2x

We then have u'(x) = - (3 sin x - 2 sin 2x, whence u'(x^) = 0 if Xj = 0, x2 = JT or cos Xj = Since u '(x) = -f3cos x ~ 4 cos 2x, therefore:

a) if 0 < (} < 4 , then umin * u ( X j ) ,

b) if (3 > 4, then umin = u(OT).

Finally, in virtue of the above and (1.8), we obtain

( 1 . 1 0 ) V * = -2A M 2A | (1. ♦ fig-) if 0 < p S 4 , Ajji > M 2(1 - (3) if (3>4, A* > 0 3

Let us substitute the values determined in (1.10) into (1.6). After some simple transformations we shall get

(1,1 1) TC(w)*

mV

2A3 (w M) 2 [ w2 + (|3 - 2)Mw + M 2] if [3> 4 M5w2

We shall next examine the function W ( z ) in the case under consideration. From the properties of this function - mentioned at the beginning of the section - and from formula (1.7) it fol­ lows that it can only be the form

(1.1 2) or (1.13) 7l(z) 2 A£(z - € ) 2 [ z2 + s (r + ¿)z + l] Mz' E, 6 * ±1, 0 < r s 1 1 l ( z) 2A|(z - a ) 2 (z -3) 2 Mz2 Itft = 1,

d i

±1

Since the function 7l(z) is non-negative on the circle |z therefore, in the case of equation (1.1 2), the inequality

TC(eiy) = —j^=(cos y - E) [2 cos y + 5(r + -p)] 2 0, 0 S y < 2JI

should hold, whencs

(1.14) £ = 1 and S = -1

3 2 By comparing the coefficients at the same powers z and z in the numerators of formulae (1.7) and (1.12), we obtain

(1.15) = -2C + 6(r + £)

(1 .16) 2 , 2 € 5 ( r ♦ !)

A5

Case (1.14) is not possible because p > 0 and the other, i.e. (1.14'), remains valid. Then, by (1.15),

(1.17) r + r" 1 = $- - 2

Equality (1.17) is possible only for p ¡i 4M. Hence, in view of (1.12), we get the right-hand side of equation C, in juxtaposi­ tion with (1.11), gives us equation C.

A suitable formula for H(F*) follows then from (1.14 ) (1.16), (1.17), and (1.10).

Proceeding analogously when the function ÎI (z) is defined by formula (1.13), we have (1.18) * -4 c o s x where x = Arg d ( 1 . 1 » , 2 * 2 COS X A * 2 whence . 2 2AS(z2 + Jn z + 1) 7t( z) = — - - - -2^ -Mz

From (1.11) it follows that equation (1.3) is of form A when 0 < ¡3 s 4, and of form B if 4<(3 < 4 M . Whereas from (1.19), (1.18) and (1.10) we obtain suitable formulae (1.9),

Proceeding analogously in case It, i.e. when AJJ < 0, we get forms D, E and F of equation (1.3) as well as suitable formulae for H(F*) defined in (1.9).

So, we have shown that equation (1.3) - with appropriateness to the values of A£ and p - takes one of the six forms. We have also expressed the upper bound of the functional H(F) by means of P, A£ and M. Consequently, it is necessary to determine the

unknown quantities A£, (3 by M and to find the intervals of va­ riability of M. For the purpose, we shall next integrate each particular differential-functional equation A - F in order to ob­ tain equations for the unknown quantities and other auxiliary para^ meters, and carry out an appropriate discussion.

2. Integration of equations for extremal functions

Let us successively consider each particular equation: 1. Equation A. In this case - by lemma 1 - we have

. w2 + J-|3Mw + M2 z2 + z 1 ( 2 . 1 ) 2 --« £ ----

---w M w z

e = ±1

Since F*(o) = 0, F*'(o) = 1, therefore £ = 1.

Integrating hoth sides of equation (2.1) in any simply connect­ ed sets which do not contain zero arid are contained in the discs ! wI < M and lz| < 1, respectively, we have

(2.2) , ♦ $ log f - = z - f ♦ C

where the branch of the logarithm is so chosen that, for z = 0, it takes the valua 0, and C is a constant.

Since on the circle Izj * 1 there is an arc \ which is trans­ formed by the function w = F*(z) onto an arc f ' of the circle I wI * = M (cf. [1]), therefore, after substituting z = e £-y and, re­ spectively, w = Me iy£T' in equation (2.2), we get

(2.3) re C * log M

Expanding the left-hand side of (2.2) in a power series in a neighbourhood of the point z = 0 and comparing the absolute terms and the coefficients at z, we have

(2.5) ~2 + A A2 +

A5

* A2 2 " 1

Making use of the fact that A| is real and of (2.3), (2.4), we obtain

(2.6) A* « ^ log M

After determining the A^ from (1.9, A) and after substituting it, together with (2.6), into (2.5), we get

(2.7) (32 log M (1 - log M) = j(M2 - 1)

This equality makes sense only for M £ (1, e). After taking account of the fact that (3 > 0, from (2.7) we have

( 2 -7 ' \ = 3 v T \ / W mU . - \ ~ o g ' W

Let us substitute the above relation into (2.6) and, next, (2.6) into (1.9, A). We shall then obtain the formula for H(F*) in case A.

From the condition p <; 4 we infer that ( 2 . 7 ) can hold only for M £ (1, M q> where MQ is the only root of the equation

(2.0) 12 log M(1 - log M) + 1 - M2 * 0 We have thus proved

Lemma 2. If in the family SR (M), M £ (1, MQ >, there is an ex­ tremal function w * F*(z) satisfying equation A, then it fulfils the equation

(2.9) ^ iog | - i x z - I + J L log M ana the equality

(2.1°) H(F* ) = l°-"T6g”"R

takes place, where MQ is the only root of equation (2.8). For M > > Mq , there is no extremal function satisfying equation A.

2 (2.1 1) fZw\2 . <»L-*Jc3. M*w2 (z2 ♦ ^ z ♦ I)2 z2 9 where -r (G < t < M) is one of the roots of the equation w*" + + (¡3 - 2) Mw + M 2 = 0.

From [6] (P- 660) and from (2.11) it follows that the point w = - 1 must be a boundary point of the domain F*(E), or else, the right-hand side of (2.11) would have a root at some interior point of the disc E. Since F*(o) = 0, F*'(o) = 1,. therefore, by (2.11), we have (w + M ) ( (2.12) zw' M 2 F + ■ * * y ; r g M2w z2 * A z + 1

the branch of the root

(2.13) P(w) =

being so chosen that, for w = 0, it takes the value

Integrating both sides of equation (2.12), after using (2.13) and after some simple transformations, we have

( 2 1 4) <M ♦ T ) 2 i o n T-. MP(w) Cm + ~ t)2 tog _ 2M2 t T + MP(w) 2M2 1 1 * P(w) _ M2 - T2 # P(w) _ M2 - T 2 p(w) a

■ z * ft' lo9 z " Y + c

where the branch of the root is chosen as above, the branches of the logarithms - so that log 1 = 0 .

i j

Since there exists a point zQ * e , 8 6 <0, 231) such that lim F*(z) = <r [6], from equation (2.14) we have

z-z„ zeE

(2.15) re {C> = 0

Let us expand the left-hand side of (2.14) in a power series in a neighbourhood of the point z = 0 and compare the absolute terms. Making use of (2.15), we get

A lQ9 jjf AJ - f t + f ■•■0 From the above relation we have

u . H > * 5 , , * ® ^

Substituting (2.16) into (1.9, B), we obtain H(F*) depending only on (3. In order to determine the H(F*), we have to determine the (3. With that end in view, let us compare the coefficients at z ajter expanding the left-hand side of equation (2.14) in a Tay­ lor series in a neighbourhood of zero; we shall then get

(2.17) f t [a j - J 3

After determining the A^ from (1.9, B) and after substituting it into (2.17) we have (2;IB) A* -16(3 - 16 (M ♦ 1) 2 4M * 48M2 or iL .¡v ♦ 16 P - 16 (M2 7 7 ? 48M2

Juxtaposing (2.16) with (2.18), ( 2 . 1 8 ) , we obtain two possi-. ble equations which should be satisfied by the (3 in the case con­ sidered:

0 - * 9 , „ n fi 0 , P * “ f ’- 16(i|2 * » (2.19) i-jp- - IRlog Jjy - m * -yI— - - -

---48M or

(2 19 ' ) . J L l0Q J L , J L /a2 ♦ 160 - 16(M2 + 1)’

K t . i y ) 2M 2M lo g 4M 4M --w ^ ^ 5 “

where 4<{J < 4 M .

At present, we shall h' concerned with the problem of the exi­ stence of roots of the above equations, as well as with the unique­ ness of solutions.

Oenote

(2.20) 4R = X and M" = T

From the conditions imposed upon (3 and M we obtain (2.21) T < A < 1 and 0 < T < 1

Let us substitute (2.20) into equations (2.19) and (2.19') we shall then get

(2.22) A( 1 2 log A) - 2T = y S 3 L l L z J ™ = _ l

(2.22') Ail - 2 log X) - 2T * - i...AI.z . j J X

On purpose to shorten the notation, let us denote

2 2 *”"V' K(A, T) » A(1 - 2 log A) - 2f, G (A, T) -- £ .1

3 A, = {(À, T ) , 0 < A S — , 0 < T sA. }

A, = {(X, T),-i-< A, < 1, 0 < T < T(X)} V F

A, * {(X, t > ~ < X < 1, T a T(X)}

A* = {(X,T) < X < 1 , T(X) < T £ X }

Ve

A 5 = {(X, T), j < X < l , TX(X) < T £ x } where T(X) = \ X(1 - 2 logX), while T ^ X ) * 2X - y ^ X 2 - 1

After examining the signs of the values of K(X, T) and G(X,T) we acquire the following information:

K(X, T) > 0 <=> (X, T) e ( A ^ A g ) K(X, T) = 0 <s»(X, T) e A 3

K(X, T) < 0 o (X, T) e Aa

G(X, T) = 0 O T * Tx(X), X € < | , 1) G(X, T) > 0 o (X, T) e A 5

By the information and the notation adopted above.it remains to consider two possibilities

(2.23) K(X, T) = G(X, T), (X. D e Aj a CAj u A 2 ) (2.23') K(X, T) = -G(X, T), (X, T) 6 A 5 n A4

In Fig. 1 we shall present domain in which (X, T) may vary, and in this domain we shall sketch the graphs of the functions

T(X), Tj(X) and T2 (X) where T2(X) = 2X +-/5\2 - I

Denote k(T) * K(X, T) and g(T) = G(X, T), where X is fixed, X G ( j , 1), while T is variable, T e (T^iXJ^X).

After examining the functions k(T) and g(T) we obtain:

I. For each equation (2.23) has one solution ^ =

Ve

-T=T2U1

II. For each — < A < 1 , Ve

T4U ) e (Tj<A), T(A)).

III. For each— < A < 1 , equation (2.23') has Ve

T5(X)

Fig. 1

equation (2.23) has one solution T

one solution T -v./w e (T(A), A) if and only if k(A) £ -g(X).

IV. For equation (2.23') has no solutions belonging to the interval (T(A),A).

After examining the functions kj^(X) = k(A) and gj(A) s g(A.) for A e (i . — > and the functions k.(A) = k(X) and g2 ( X ) =

2 Ve

-g(A) for Ae(^, 1) we get

Lemmai 3. For each X € <X0, equation (2.23) T = Tj(X) e ( T ^ A ) , A), where AQ e (¿, -i) solution

root of the equation

has exactly one is the only

II. For eachA€~-, l), equation (2.23) possesses exactly one solution T = T4 U ) 6 (T^(A), T(A)).

III. For each pair (X, T) e A 5 o A 4 , equation (2.23) possesses no solutions.

It remains to decide whether, for distinct A € < A 0, 1), the roots of equation (2.23) are distinct and, if A 6 < X 0, 1), then in what interval the values of Tj(A) and T^(A) vary.

After some simple calculations we get that the functions Tj(A), T^(A) are decreasing functions of variable A in suitable intervals, i.e. that the function

T*(A)

t3(a), a ^ x < ; ^

T4(A), yg, S A < 1

decreases from the value AQ - TQ e (^, ) to

From lemma 3 and the above considerations there follows:

Lemma 4. For each T e <j|, To >, equation (2.22) possesses ex­ actly one solution X * \(T) 6 < T Q , 1) where TQ *

the only root of the equation

For T e (0, 1), Tc > equation (2.22) has no solution belonging to the interval (T, 1). If T = j-fi tben (A)T = 1, whe­ reas if T * T , then A(Ta ) * TQ ; these are limit cases.

For T e (0., 1) and X e ( T , 1), equation (2.22') has no solu­ tions.

Corollary 1. For each M e < M Q , equation (2.19) possesses exactly one solution belonging to the interval (4, 4M) where MQ is the only root of equation (2.10). For A > 1, equa­ tion (2.19) has no solutions from the interval (4, 4M).

II. For each M > 1, (3 e (4, 4M) equation (2.19') possesses no solutions.

We have thus shown

Lemma 5. If in the family SR (M), M > 1, there is an extre­ mal function w = F*(z) satisfying equation B, then

H(F*) • ~ ~ - — log + -^ (~- - (3 + 1) > 2M 2M 4M L M B

where 3 is the only root of equation (2.19), belonging to the

in-O i i

terval (4, 4M), while M 6 (Ml)I a j) where Mq is the onlv root of equation (2.10). Besides, the extremal function w » F*(z) sati­ sfies (2.14) where

0 (3

C = -5m loo + A5 - + F

For M £ (Mq , --j) , there is no extremal function satisfying e- quation B.

3. Equation C. In thi3 case, the equation can be written in the form

(2.24) / 2W \2 (w ♦ M >2 (w + # (w *

A w'; MTw2

(Z + l)2 (z + £>) (z + ~ )

2

where -T(0 < T < M) is one of the roots of the equation w + + (0 - 2 )Mw + M 2 = 0, while -¿>(0 < £ s i ) is one of the roots of

the equation z2 + (|j- - 2)z + 1 = 0,

Note that in the domain {• * E - (-1, - £ > there is a single- -valued branch p(z) From (2.24) it follows that

-1

= * F*

frgi)

since F*(-g) * 0. Consequently,

\n

the domain F*(§) there is a single-valued branch of the P(w) defined in (2.13). Let us adopt p(o) = £, P(o) *

Since F»(o) « 0, F*"(o) = 1, equation (2.24) can be repre­ sented for z e

i

in the form

Integrating equation (2.25), we obtain ufter 3ome simple trans­ formations (2.26) log . Ü L + J ê Z log -2M T T + MP(w) 2M T 1 ♦ P(w.) . M2 - T 2 . __ P(w) _ M2 - T 2 . P(w) „ M M 2P5 (w) - T ? P2 (w) - 1

« i l J L S ù l log - (: .rjel! log

2g ¿p + p(z) 2# 1 + p(z)

+ c

P <z)-/p £> P <z) - 1

where C is a constant, and P(w), p(z) are the branches of the roots, chosen before, while the branch of the logarithm is so chosen that log 1 » 0.

Since, with z- -£ w - -z, therefore, passing to the limit of z - -£ in (2.26), we have

C = 0

Next, expanding both sides of equation (2.26) in a power se­ ries in a neighbourhood of z = 0, after comparing the absolute

M X 1 P

terms and making use of the relations + = P " 2, g + _. » .p. _ 2) we get

(2.27) A* = 2(1 - ^-)

Let us make another use of the expansion of equation (2.26) in a power series in question and compare the coefficients at z. We then obtain

« ■ 28> â [ * 2 - JT r j ' ^ 2 * A5 - * 5 i *

* z i ■ t i ■ é (1 ■ A ’ * A • 1 * " S

Determining the A^ from (1.9, C) and substituting it, along, with (2.27), into (2.28), we have

11M - 13 ¡3--- 2__

Solving the inequality ¡3a4M, we deduce that M £. We have thus proved

Lemma 6. If in the family SR (M)-, M 2: there is an extre­ mal function w = F*(z) satisfying equation C, then • A j £ P g (M ) • • P,(M) where P0 (M), P,(M) are defined in the introduction. This estimation is true for M a For M < there is no extremal function satisfying equation C.

Consider the function w = 9(z, M), holomorphic and univalent in |z| < 1, defined by equation (0.5). Since £P (z , M) = z +

+ 2(1 - ^ ) z 2 + (3 " 'S' * * •••* therefore ^(z, M) e SR (M) and M

H($>) = H(F*). It can easily be noticed that the function satisfies equation (2.26).

4. Equation D. In this case, the equation can be written in the form 9 M2 -2 (w + M) (w - T) (w - ~ r ) (2.2S) --- - r T .. .... " M w (z ♦ n 2 (z - ¿.Hz - h .... ... .. "2- - - -ÙL~ z o

where % e (0, M) is one of the roots of the equation w + M(p -2)w + + M2 = 0, while g e (0, 1) is one of the roots of the equation z2 ♦ (Jj- - 2)z ♦ 1 = 0.

Proceeding similarly as in case C, we have A* - 2(1 - ¿) > 0

whence we get a contradiction since, in the case under considera­ tion, A* < 0.

So, we have shown

Lemma 7. There is no extremal function with respect tc the fun­ ctional H(F), satisfying equation 0.

5. Equation E. In this case, the equation after some transfor­ mations takes the form

(2.30) 2 !L w j — jj! » - i

Inr . itlng both sides of the above equation, we have

(2.31) ^ . i = 2 , y . c

where C is a constant.

Let us expand the left-hand side of (2.31) in a power series in a neighbourhood of z = 0 and compare the absolute terms as well as the coefficients at z. We shall then get

(2.32) ' C * -Aij;

(2.32') 4 ? + A * 2- A$ = 1 M

Determining the A5 from (1.9, £) and substituting it into (2.32'), we obtain

a 22 " ¥ '

-Hence, after taking account of the sign of AJ (AJ < 0 ) , we get *2 3 ‘ f a '

and, thereby, we have determined the H(F*) in the case consider­ ed. 8y examining equation (2.31) for C one can

easi-li

ly demonstrate that it possesses a solution belonging to the family Sr(M) only for M £

Mo have thus proved

Lemma 8. If in the family SR (M), M > 1, ther« Is sn extre­ mal function w = F*(z) satisfying equation E, then

I S I E P ®

and moreover, this function satisfies the equation

^ 1 1 s z + i + y ^ ( l --ij)'

This case can hold only for M 2 jj.

/zw\2 iw * M) (w + <w + T*) (2.33) ( ¿ a - ) - - - r~=--- =

w M w

(z - l)2 (z p)(z +

--- ? — *

-Proceeding similarly as in case C, we get A 2 = '2(1 " TT*

Let us substitute the above relations into (1.9, F then obtain H(F*) = -P2(M) • P3(M) where P2(M), Pj(M) in the introduction.

From the inequality /3 > 0 follows that M < ^ j - Consequently, we have proved

Lemma 9. If in the family SR (M), M > 1, there mal function satisfying equation F, then

H(F*) .» -P2 (M) • P3(M) and moreover, this function satisfies the equation

(2.34) Â lo® f i ' S l î i " & l09 I ~ r Î ( î 8 + . M 2 ». T 2 P(w) T O T T ? " . M . „PCy), . a l0B

M <r

P2(w) - l

2M

e *

P(z)

-

&

log * - P (,> ♦ (l - A ■■■■ K » 1 - 2M 1 * p(z)

e

P2<z) -

p‘

i - ¿>2 P(z) £ p2 (z) - 1 ); we shall are defined is an

extre-Thi3 case can hold only for M < yr.

Consider the function w = -«PC-z, M) where w = “PC z, M) is defined by equation (0.5). Since-9(-z, M) = z 2(1 - -^)z2 + + ( 3 - |j- + -^-j)z3 + ..., therefore-<P (-z, M) € SR (M) and HC-i’t-z.M))

M

= H(F*). It is easy to demonstrate that the function w =-<P(-z, M) satisfies equation (2.34).

3. The fundamental theorem

In the investigations made so far, we have considered all pos­ sible forms of differential-functional equation (1.3). At present, we shall proceed to proving

The fundamental theorem

2 3

For any function w = F(z) = z + A2z + A?z ♦ ly SR (M), M > 1, the following sharp estimation (3.1)—(3.5) -2(1 - ¿ ) 2(3 _{M )} A , £

WTV(

“ - y - ir> of the fami-if 1 < M si13₁₁ if V e S M S H if Mq S M if M > 13

takes place; where in (3.3), (3.4) Mq is the only root of the equation 12 log M(1 - log M) + 1 - M 2 = 0, while the |3 occuring in (3.4) is the only root of the equation

a

a -

4 _

2M 2M 109 4R'- JL ^ * i6<* ~ 16(m2 _4BM *■ (belonging to the interval (4, 4M)),

P r o o f . In lemmas 2,5, and 6 , 8 , 9 we have obtained the formulae for the values H(F*) of functional (1.1), as well as the intervals in which suitahle estimations can hold true. Since the intervals obtained are not disjoint, the proof of the theorem will consist in determining the sharp estimations in the common parts of the above-mentioned intervals.

Note first that, in the interval (1, ~ ~ > formulae (3.1) or (3.3) can hold (see lemmas 9, 2). Comparing both values of the functional, we arrive at the conclusion that the estimation valid in this interval is (3.1); the function w =-<J> (-z, M), where w = = P(z, M) is a Pick function defined by equation (0.5), realizes the equality in estimation (3 1) and belongs to the family SD (M).

IX K

Similarly, in the interval < n > V e ‘> there can hold two cases, i.e. (3.2) or (3.3) (see lemmas 2, 8). Comparing both values of the functional in the interval considered, we infer ¿.hat the 'es­

t im a t io n valid is (3.2), equality taking place for the function w* = F*(z) defined by equation (2.30), This function maps the disc E onto the disc |w I < M from which two arcs lying on the real axis and issuing from the points Wj 2 = ~M have been removed.

Let us next consider the interval <Ve', M0 > where Mq is defined in theorem 1. In this interval', formulae (3.2) or (3.3) can take place. After comparing suitabale values of the functional H(F) in the interval under consideration we deduce that the estimation hol­ ding true is (3.3); the equality in estimation (3.3) is realized by the function w s F*(z) defined by equation (2.9); the proof of the existence of such function of the family Sp(M) can be found in

CO-Let us now take into consideration the interval < M Q , i|>. In this interval, estimations (3.4) or (3.2) can hold. Comparing both values of the functional in the interval considered, we infer that (3.4) is valid; the equality in estimation (3.4) is realized by the function w = F*(z) defined by equation (2.14); the proof of the existence of such function of the family SR (M) can be found in paper [4].

What is left for us is to consider the interval <^-j, +0°) • From lemmas 6 and 8 it follows that, in the interval under considera­ tion, formulae (3.5) or (3.2) can hold true. From the compari­ son of both values of the functional we conclude that formula (3.5)

is valid in this interval; the function realizing the equality in this estimation is w * <P(z, M) defined by equation (0.5) which, as we know, belongs to the family Sp(M). Consequently, the theorem has been proved.

2 ^

Corrollary 2. For any function w * F(z) = z + A ?z + A }z * . . , of the family SR (M), M > I, the product A2 • A ? is greater than or equal to suitable values (3.1)-(3.5) multiplied by minus one.

This corollary results from the fact that if F € Sft(M),then the function w = -F(-z), too, belongs 1;o tho family SR (M):

References

[ l j D z i u b i r t s k i I., L equation des fonctions extrema- les dans la famille des fonctions univalentes bornees et syrome- triques, Pr. Wydz. mat.-przyr. Łódź TN. 1960, nr 65.

[ 2 ] J a k u b o w s k i Z . J . , Z i e l i r t s k a A., Z y- a k o w s k a K., Sharp estimation of even coefficients of bounded symmetric univalent functions. Ann. Polon. Math. (to appear).

[3] J a n o w s k i W., Le maximum des coefficients A^ et A-j des fonctions univalentes bornees, Ann. Polon. Math. 1955, t. II, vol. 2, p. 145-160.

[4] M i k o ł a j c z y k L., Domaine de variation des coeffi­ cients A 2 et Aj des fonctions univalentes bornees a coeffi­ cients reels, Ann. Polon. Math, 1967, t. XIX. p. 81-106.

[5] P i c k G., Über die konforme Abbildung eines Kreises auf ein schlichtes und zugleich beschranktes Gebiet, Sitzgsber. Kaiserl. Akad. Wiss^ Wien, Abt. II a, 1917, vol. 126, p. 247-263. [6] R o y d e n H. L., The coefficients problem for bounded

schlicht functions, Proc. NAS 1949, vol. 35, p, 657-662. [ 7 ] S c h a e f f e r A . C . , S p e n c e r D . C . , The coef­

ficients of schlicht functions, Duke Math. 3 . 1945, vol. 12. [8] S c h i f f e t M., T a m m i 0., The fourth coeffi­

cient of bounded real univalent functions, Ann. Acad. Sei. Fen-* nicae 1965, eer. A, t. I. vol, 354, p. 1-34.

of bounded schiicht functions, Ann. Scl. . Fannicae 1953. ser. A, t. I f vol. 149.

[10] Z i e l i ń s k a A., Z y s k o w s k a K., Estimation of the s i x t h coefficient in the class of univalent bounded functions with real coefficient, Aon. Polon. Math, (to appear). [11] Z i e l i ń s k a A., Z y s k o w s k a K., Estimation

of the eighth coefficient of bounded univalent functions with real coefficients, Oemonstratio Math. 1979, vol. XII, no 4, p. 231-246.

Institute of Mathematics The University of Łódź

Longin Pietrasik

OSZACOWANIE FUNKCJONAŁU A2 • A 3

W KLASIE FUNKCJI OGRANICZONYCH, SYMETRYCZNYCH I JEDNORODNYCH

Oznaczmy przez 5R(M), M > 1, rodzinę funkcji F(z) * z + holomorfi-n*2

cznych i jednolistnych w kole E = { 2 : |z| < 1}, o rzeczywistych współczynni­ kach i spełniających warunek I F(z)l S M , z e E.

W pracy uzyskano dokładne oszacowanie funkcjonału H(F) = A,* A, w klasach SR(M), M > 1.